Long-range-interacting topological photonic lattices breaking channel-bandwidth limit

The presence of long-range interactions is crucial in distinguishing between abstract complex networks and wave systems. In photonics, because electromagnetic interactions between optical elements generally decay rapidly with spatial distance, most wave phenomena are modeled with neighboring interactions, which account for only a small part of conceptually possible networks. Here, we explore the impact of substantial long-range interactions in topological photonics. We demonstrate that a crystalline structure, characterized by long-range interactions in the absence of neighboring ones, can be interpreted as an overlapped lattice. This overlap model facilitates the realization of higher values of topological invariants while maintaining bandgap width in photonic topological insulators. This breaking of topology-bandgap tradeoff enables topologically protected multichannel signal processing with broad bandwidths. Under practically accessible system parameters, the result paves the way to the extension of topological physics to network science.


Note S1. Tight-binding description of waveguide-loop coupling
We review the photonic tight-binding formulation of the ring resonators coupled via non-resonant waveguide loops (Fig. S1), which has been widely employed in topological photonics 1,2 .We consider only the pseudospin up (counterclockwise wave circulation) modes, neglecting the interactions between opposite pseudospins.The temporal coupled-mode theory (CMT) model 1,2 for Fig. S1 is: where ψa and ψb represent the fields in each resonator, s1, s2, s3, and s4 denote the propagating fields at each position of the waveguide loop (Fig. S1), ω0 is the resonant frequency of both resonators, τ is the lifetime of the resonance modes denoting the external energy leakage to the waveguide loop, and φt and φb are the phase changes of the light along the top and bottom parts of the waveguide loop, respectively.From Eq. (S1), we obtain where H is the Hamiltonian of the system, a † , b † , a, and b are the creation and annihilation operators of each resonator, respectively, and t = 1/τ denotes the coupling strength.
Extending the two-resonator system to a lattice shown in Fig. 1a in the main text is straightforward.A lattice of resonators is constructed by connecting the ring resonators via the non-resonant waveguide loops with the tailored offset δ (Fig. S1) to achieve the phase difference 2φ between the loop arms.

Note S2. Chern number illustration in the Hofstadter butterfly
The colored Hofstadter butterflies in Fig. 1c, 2d, and 2e of the main text illustrate the band structures and the gap Chern numbers of the Hamiltonian in Eq. ( 1) in the main text.We color the butterfly pixelwise: 1440 × 1920 pixels for the αand ω-axes, respectively.Black and pastelcolored pixels correspond to the α-ω states in the bands and the gaps, respectively.
To determine the color of each point in the gaps, we calculate the ranges of the bands for each α = p/1440 with integer p between 0 and 1440.Substituting the gauge of Eq. ( 2) in the main text into the Hamiltonian, we obtain ( ) where the integrand H1(ky) diagonalizes H.The q-periodicity of H1(ky) due to the rationality of α = p/q allows another Fourier series along the x-axis: with the truncated index j = 1, 2, …, q.We finally obtain the finite-dimensional q-band Hamiltonians for the reciprocal vector k = (kx, ky) in the magnetic Brillouin zone: where we identify q+1 with 1 for the index j.The finite-dimensional Hamiltonian H2(kx, ky) is diagonalized to determine the eigenfrequencies at each k point.
We show an examplary band structure under α = 1/4 for the k points in the magnetic Brillouin zone (black box in Fig. S2), which consists of four repeated regions (a unit red region in Fig. S2).The symmetry of the Hamiltonian in Eq. (S9) results in q repetitions of the band structures inside the magnetic Brillouin zone for α = p/q, allowing the ranges of kx and ky values between 0 and 2π/q to fully determine each band 3 .Therefore, we discretize the 2π/q × 2π/q square plaquette (the repeated part in Fig. S2) into 20 × 20 points in kx and ky directions to calculate the eigenfrequencies.The range of each band is then obtained from the extremums among the 400 frequency values obtained at the k points.
After quantifying the range of the bands, we determine whether the frequency corresponding to each pixel belongs to a band or not.For the frequencies in bandgaps, we assign the pixel the color corresponding to the gap Chern number calculated from the TKNN formula: Eq. (3) in the main text.

Note S3. Butterfly wing area calculation
As the performance figure of signal transport, we numerically calculate each wing's area of the Hofstadter butterfly.We approximate the wing geometry to the hexacontagon (or 60-gon) defined with 60 vertices sampled from the wing boundary.

Note S4. System parameters of the overlapped lattice for silicon photonics
We suggest a silicon-on-chip implementation of the topological overlapped lattice by extracting the tight-binding parameters, including coupling strength t, cross-coupling κ', and system loss κext, from the full-wave numerical analysis using the finite-difference time domain (FDTD) method 4 .
From the results, we verify that κ'/t is sufficiently suppressed by adopting a conventional silicon waveguide crossing design 5 , which demonstrates the validity of the ideal tight-binding modelling for the lattice overlap.Table S1 summarizes the proposed geometry and the coupling parameters.
As a practical implementation, we consider the silicon photonic slab structure, assuming the quasi-transverse electric (TE) mode operation at the telecom wavelength λ0 = 1550 nm.To realize the lattice overlap, we employ the low-loss and low-crosstalk waveguide crossing design 5 to the junctions between lattices, where the simulation geometry and result are shown in Fig. S4.
Near the target wavelength λ0, the insertion loss and crosstalk are less than 0.15 dB and −50 dB, respectively.To investigate the impact of the crosstalk on the lattice overlap, we examine the periodic system shown in Fig. S5a.The system is modeled by the temporal CMT with the coupling strength t (solid lines in Fig. S5a), unwanted crosstalk coupling κ' (dashed lines in Fig. S5a), and internal loss κext of each resonator.Figure S5b shows the implementation of the system's building block using ring resonators and zero-field waveguided couplings, where two waveguide loops cross each other at four points.The effective CMT model parameters can be extracted from the FDTD simulation of the geometry in Fig. S5b, as follows.
We first solve the tight-binding Hamiltonian of the CMT model to obtain an implicit dispersion relation.The model extends along the x and y directions, resulting in the two-band Hamiltonian: , , ) where an,m and bn,m are the resonator fields (Fig. S5a) and ω0 is the resonance frequency.We apply the Bloch theorem to obtain and where ξx and ξy are the eigenvalues of the translation operators along the x and y directions, respectively.The Hamiltonian then becomes (1 )(1 ) ( ) We focus on the case where ξ = ξx = ξy to simplify the problem.Using the harmonic condition with the system frequency ω, we obtain The Hadamard basis ,, ,, diagonalizes the system, providing the relation between ξ and Δ = ω -ω0 for each band: ) 0, 0 where ξc and ξd are the ξ values corresponding to each band.The equations are equivalent to which are the desired implicit dispersion relations.S1.
To numerically calculate the implicit dispersion relations, we perform the FDTD simulation on the coupling region in the blue region of Fig. S6.The simulation results in an 8 × 8 scattering matrix Sij(ω), where each input field ψj and output field φi are related by φi = ΣjSijψj (Fig. S6).We transform the scattering matrix to the transfer matrix Mc, defined as where T denotes the transpose of the vector.To construct the full transfer matrix through a unit cell of the periodic structure, the propagation through the part of resonators should be considered, as follows: where diag(v) is a diagonal matrix from the elements of v and T is the propagation time determined by the length of the resonator.The full transfer matrix is then obtained as The numerical dispersion relation is achieved by the following generalized eigenproblem: ( ) diag( , , , , , , , ) 0 Same as the simplification in the CMT model, we assume ξ = ξx = ξy.Moreover, we employ the Hadamard basis in Eq. (S15) by applying the following basis transformation matrix V: 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 2 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 where the transform matrix in the new basis becomes M H (ω) = V † M(ω)V.Neglecting the crosstalk between the modes, M H (ω) is a block diagonal matrix where Mc H (ω) and Md H (ω) are 4 × 4 matrices corresponding to the basis cn,m and dn,m in Eq. (S15).
We note that   (S17).To compare those equations, we determine the resonator length by changing the value of T for the resonance near 1550 nm (λ0 = 1551.5nm) with the free spectral range around 500 GHz.

Note S5. Overlapping disparate lattices
In the main text, we have focused on N-folded overlaps of identical lattices, which allow for the N multiplications of the gap Chern number C. By overlapping disparate lattices, we can design an Nfolded lattice possessing an arbitrary C other than the multiples of N. Figure S8 shows a 3-folded lattice overlap, which consists of the lattices with two different magnetic fluxes α.We employ the α values (black dotted arrows in Fig. S8b) to support 4 edge modes in total at the operation frequency range (white dotted arrows in Fig. S8b), which is determined by the narrowest band gap among the three gaps of interest.To confirm the design, we examine the band structure of onedimensional (1D) ribbon geometry of the domain (Fig. S9).To clearly reveal the edge modes resolving the degeneracy, we apply random perturbations as described in Figs.3a,d of the main text.Figure S9b illustrates the 4 forward-propagating edge modes at the operation frequency range as expected.
In general, to construct the domain possessing the gap Chern number M by overlapping N disparate lattices, we require at least one lattice to have the gap Chern number no less than K = ⌈M/N⌉ according to the pigeonhole principle.Therefore, the lattice with the gap Chern number K imposes the limit on the signal transport performance according to its corresponding bandgap.

Note S6. Calculating band structures using ribbon geometries
To numerically calculate the band structures in Fig. 3a-c in the main text and Figs.S9 and S11, we employ the ribbon geometry: the finite system size along the y-axis while maintaining the translation symmetry along the x-axis.
We consider the geometry and hopping phases in the connections between the resonators to determine the minimum supercell that repeats in the structure.Under the phase-coded representation illustrated in Fig. S10a, Fig. S10b,c shows two exemplary designs.To minimize the x-axis size of the supercell, we employ the gauge varying along the y axis, while the gap Chern number remains unchanged under the gauge transformation.Based on the constructed supercell, we apply the Bloch boundary condition to numerically solve the finite Hamiltonian and obtain the band structure.In Fig. S11, we provide the band structure of each port shown in Fig. 4b of the main text using the ribbon geometries.The full eigenvalue equation of the system including the scattering region is † 22 † 11 † 00 00 0 0 00 where VC connects the port and the scattering region, HS is the Hamiltonian of the scattering region, and the eigenmode consists of the finite-dimensional vectors ψn for the integer n ≥ 0 (Fig. S12).
The equations of the lowest two rows are † 0 1 0 † 1 2 1 0 Cancelling out ψ0, we obtain ( ) ( ) where I is the identity matrix.The other equations obtained from the upper rows of the matrix equation Eq. (S27) restrict ψ1 and ψ2 to be the linear combinations of the eigenmodes supported by the port: out out decay decay in in 1 0, 0, 0, out decay in 2 0, 0, 0, where ψP0,j in , ψP0,i out , ψP0,k decay , λj in , λi out , and λk decay are the eigenvectors and eigenvalues obtained from Eq. (S25), corresponding to the propagating inward, outward, and decaying solutions according to the indices, and Sij and Uij are the scattering parameters from the jth input to the ith propagating and evanescent modes of the port, which are unknown linear coefficients to be determined.We note that Sij is obtained by plugging Eq. (S30) into Eq.(S28).The transmission for the incoherent incidence against hopping-phase disorder with different Uphase.
For each type of disorder with varying Udiag and Uphase, 50 random realizations are estimated, where each line in a and b denotes a realization.The shaded regions denote the frequency ranges of perfect beam splitting operation against disorder.

Note S9. Performance limitations in lattice sizes and bandwidths
According to the tight-binding model in Eq. ( 1) in the main text, the device bandwidth using our lattice overlap model is primarily governed by the hopping strength t.This ideal condition is valid because the value of t (= 40 GHz) is much smaller than the signal bandwidths of other optical elements, such as adiabatic waveguide crossing (Fig. S4b).The dependency of the bandwidths on hopping strength is illustrated in Fig. S14, comparing the conventional Hofstadter lattice (Fig. S14a,b,e,f) and our overlapped lattice (Fig. S14c,d,g,h) for different values of hopping strengths t/2 and t.As demonstrated, the bandwidth is directionally proportional to the hopping strength, at least, in an ideal condition.However, in our lattice overlap model, the performance is limited due to the system size issue.Figure S15 shows the relationship between the number of lattice overlaps N and the characteristic length scale d: the distance between the resonator connected through a waveguide coupler.As shown in Fig. S15a,b, the averaged characteristic length increases proportionally to ~N1/2 even when neglecting the spatial length of the adiabatic waveguide crossing structure.In a practical realization, this footprint issues becomes worsen.We note that the entire size of the lattice is governed by waveguide crossing structures due to their tapering shapes for wide operation bandwidths.Because the number of the necessary crossing is proportional to the overlap number N, the averaged characteristic length becomes ~N.This enlarged system footprint enforces the degradation of the integration level of the lattice (Fig. S16c).Importantly, to maintain the lattice size, a more compact design of waveguide crossing structures is necessary, such as impedance matching techniques, though such approaches typically lead to the significant degradation of operation bandwidths.

Fig. S1 .
Fig. S1.Coupled mode theory for the tight-binding Hamiltonian.A schematic for the indirect coupling between two ring resonators with the tailored gauge field.δ denotes the displacement of the waveguide loop to induce the phase difference between the loop arms.
)where (m1, m2) and (n1, n2) are the pairs of the integer indices denoting the positions of the mth and nth resonators in Eq. (1) in the main text, respectively.The index for the pseudospin σ is omitted because both pseudospins are decoupled and possess identical band structure.Due to the discrete translational symmetry of the Hamiltonian along the y-axis, we diagonalize H by introducing the following Fourier series: n1.The Hamiltonian in Eq. (S6) then becomes ( )

Fig. S2 .
Fig. S2.Band structure example of the Hofstadter model for α = 1/4.The band structure is calculated using Eq.(S9) in the magnetic Brillouin zone (black box).The 4-fold repetition of band structure (red shaded region) is observed due to the magnetic translational symmetry.

Fig. S3 .
Fig. S3.Approximated wings of the Hofstadter butterfly.a, 12 wings for C = 3 according to the range of α. b, An inset for describing the nonuniform sampling of the α range (3/4, 4/5) to compose a 60-gon.

Fig. S4 .
Fig. S4.Crossing geometry for lattice overlap.a, The silicon waveguide crossing 5 and propagating quasi-TE field obtained from the FDTD simulation.b, The simulation result for the insertion loss and crosstalk.

Fig. S5 .
Fig. S5.Waveguide crossing in the lattice overlap.a, The structure for estimating the hopping strength t and crosstalk coupling κ' in the silicon photonics implementation.an,m and bn,m denote the fields of resonance modes.b, A schematic of the unit cell implementation.Dr, Dc, dg, dc, rf, lc, and tc represent the geometric parameters, where their values are provided in TableS1.
because the four eigenvalues of Mc H (ω) corresponds to forward and backward eigenvalues of the spin up and down modes, where the same holds for Md H (ω).

Fig. S6.
Fig. S6.a, Silicon photonics implementation.A schematic of silicon photonics implementation of the structure shown in Fig. S5a.M(ω) with the corresponding box denotes the transfer matrix of the unit cell.M(ω) consists of Mc(ω) and Mr(ω), which are obtained from the FDTD analysis and Eq.(S19), respectively.b, The unit cell structure presenting the propagating fields at several points.The propagating fields ψi, ψi', φi and φi' are connected by Eqs.(S18) and (S19).
) is the numerical dispersion relation based on the results of the full-wave analysis using the FDTD simulation, which is the counterpart of the analytical formulation of Eq.

Fig. S7 .
Fig. S7.System parameter extraction.The real (a) and imaginary parts (b) of the implicit dispersion relations defined in Eq. (S23).The stars denote the operation wavelength of the system (λ = 1551.5nm), where the real-values' slopes and the imaginary values are used to extract the coupling strengths t and κ'.

Fig. S8. The 3 -
Fig. S8.The 3-folded lattice overlap.a, A schematic of the 3-folded overlapped lattice with two different α and C. Colored boxes (green, blue, and red) are depicted to distinguish each lattice component.b, The operation frequency range of the overlapped lattice using a part of the Hofstadter butterfly.Each colored solid line corresponds to the colored box in a.

Fig. S10 .
Fig. S10.Supercells in the ribbon geometries.a, The unit geometry and hopping phase between the resonators.Circles represent the resonators, and φnm and φmn are the hoping phases in the interactions between the mth and nth resonators.In the below square, four hopping phases between four resonators exhibit a synthetic flux 2πα.b,c, Two examples of ribbon geometries illustrated with the supercell.The supercells exhibit different periodicity according to the phase and connectivity between the resonators.Color bar denotes the value of hopping phases.

P
Fig. S11.Band analysis of each port in Fig. 4. a-c, Schematics of the port 1 (a), 2 (b) and 3 (c) in the scattering region of Fig. 4 in the main text.While arrow directions denote the propagating directions of the edge modes, a pair of the dashed arrows represents the annihilation of the edge modes due to the counter-propagation.d-f, Corresponding band structures for a (d), b (e) and c (f). ymean denotes the y-axis center-of-mass values of the edge modes.

Fig. S12 .
Fig. S12.Calculation of the S-parameters from the tight binding model.The exemplary semiinfinite system with a single port.HPC, HS, VP, VC, and ψn are the port matrix, the Hamiltonian of the scattering region, the interconnecting matrix between the ports, the interconnecting matrix between the port and the scattering region, and the eigenmodes of the ports and the scattering region, respectively.

Fig. S13 .
Fig. S13.Beam splitting of incoherent light against diagonal and hopping-phase disorder.a, The transmittances for the incoherent incidence against diagonal disorder with different Udiag.b,

Fig. S14 .
Fig. S14.Lattice bandwidths governed by hopping constants.a-d, The lattice structures and eh, corresponding butterflies for the conventional Hofstadter model (a,b,e,f) and the lattice overlap model (c,d,g,h) with different values of hopping strength: t/2 (a,c,e,g) and t (b,d,f,h).In a-d, blue rounded squares illustrate the resonators.The insets in a-d describe the change of the gap between the resonator and waveguide coupler for controlling the evanescent coupling.The symbol 'd' in ad denotes the distance between connected resonators.

Fig. S15 .
Fig. S15.Characteristic length scales and integration.a, The characteristic length d as a function of the overlap number N. b, The schematics for illustrating the characteristic length at each lattice overlap number.The symbol 'IN' (or 'PN') represents ideal (or practical) cases, neglecting (or considering) the coupler length.In b, blue rounded squares illustrate the resonators.c, The variation of the lattice integration with respect to N. Nresonator depicts the number of resonators inside the unit cell, and A denotes the unit cell area.

Fig. S16 .
Fig. S16.Overlapping two Haldane lattices.a,b, Two Haldane lattices with nearest-neighbor (solid lines) and next-nearest-neighbor (dotted lines) interactions between the resonators (circles).b describes the deformed lattice of the Haldane lattice for satisfying the design criteria of the lattice overlap, while maintaining the original Hamiltonian.c, A possible network structure of the overlapped Haldane lattice.

Fig. S17 .
Fig. S17.Competition of NN and NNN interactions in the doubly overlapped Hofstadter lattice.a, Different strengths of NN interactions imposed on the overlapped lattice.b,c, Evolution of the band structure under increasing NN interactions b, without and c, with hopping phases.We color the band in gray and the topologically nontrivial bandgaps with colors corresponding to the gap Chern numbers C.